11.2 HyperbolasObjectives:
1. Define a hyperbola2. Write the equation of a hyperbola3. Identify important characteristics of
hyperbolas4. Graph hyperbolas
HyperbolaThe set of all points for which the difference
of the distances from two points is constant.
Equation of a Hyperbola Centered on the Origin
Characteristics of a HyperbolaImportant
Facts:
The hyperbola bends toward the foci
The positive term determines which way the hyperbola opens
The distance between the foci is 2c
The distance between the vertices is 2a
The center is the midpoint between the foci and the midpoint between the vertices
c2 = a2 + b2
Example #1Show that the graph of the equation is a
hyperbola. Graph it and its asymptotes. Find the equations of the asymptotes, and label the foci and the vertices.64164 22 xy
124
1416
64
64
64
16
64
4
2
2
2
2
22
22
xy
xy
xy
5220
416
24
2,4
22
c
c
c
ba
xy
xy
xb
ay
Asymptotes
22
4
:
Example #1Show that the graph of the equation is a
hyperbola. Graph it and its asymptotes. Find the equations of the asymptotes, and label the foci and the vertices.64164 22 xy
124
:
2
2
2
2
xy
Equation
xy
Asymptotes
2
:
52,0:
Foci
4,0:
Vertices
1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x
12345678910
–1–2–3–4–5–6–7–8–9–10
y
Example #2Graph the following hyperbola using a
graphing calculator.
2045 22 yx
4
520
4
520
5204
2
22
22
xy
xy
xy
Example #3AA. Find the equation of the hyperbola that has
vertices at (2, 0) and (-2, 0) and passes through Then sketch its graph by using the asymptotes, and label the foci.
3,4
13
4
16
13
2
4
12
2
2
2
2
2
2
2
2
2
b
b
b
yx
1
33
33
13
4
2
2
2
b
b
b
b
With the vertices on the x-axis, this implies a = 2.
5
14
12 22
c
14
22
yx
xy
xa
by
2
1
Example #3AA. Find the equation of the hyperbola that has
vertices at (2, 0) and (-2, 0) and passes through Then sketch its graph by using the asymptotes, and label the foci.
3,4
14
:
22
yx
Equation
xy
Asymptotes
2
1
:
0,5:
Foci
1 2 3 4 5 6–1–2–3–4–5–6 x
1
2
3
4
5
6
–1
–2
–3
–4
–5
–6
y
Example #3BB. Find the equation of a hyperbola with y-
intercepts at
±7 and an asymptote at xy3
1
With it intersecting the y-axis, this implies that a = 7. From the equation of the asymptote we get:
213
17
bbb
a1
217 2
2
2
2
xy
Example #3CC. Find the equation of a hyperbola with foci at
(±8, 0) and a vertex at 0,25
14
14
5064
258
8,25
2
2
22
b
b
b
b
ca
1
5064
1258
22
2
2
2
2
yx
yx
Example #4An airplane crashed and was heard by a park
ranger and by a family camping in a park. The park ranger and the family are ¼ mile apart and the ranger heard the sound 1 second before the family. The speed of sound in air is approximately 1100 feet per second. Describe the possible locations of the plane crash.The family and the ranger are placed at opposite foci of a hyperbolic curve. The crash occurred closer to the ranger than the family so the crash occurred on the branch of the hyperbola closest to the ranger.Since sound travels at 1100 ft/sec, after 1 sec it will have traveled 1100 ft. This implies the crash was 1100 ft closer to the ranger than the family, which also means the vertices are 1100 ft apart. Since 1 mile has 5280 ft, ¼ a mile is 5280 ÷ 4 = 1320 ft, which is the distance between the foci.
Example #4Describe the possible locations of the plane
crash.Distance between foci: 1320 ftDistance between vertices: 1100 ft
660
13202
550
11002
c
c
a
a
365
100,133
500,302600,435
550660
2
2
22
b
b
b
b
1365550 2
2
2
2
yx
Ranger Family
200 400 600 800 10001200–200–400–600–800–1000–1200 x
200
400
600
800
1000
1200
–200
–400
–600
–800
–1000
–1200
y
The crash occurred somewhere on the left branch of the hyperbola.